1.2.1 The Efficiency Objective—The Switch

A circuit element as simple as a light switch reminds us that the extreme requirements in power electronics are not especially novel. Ideally, when a switch is on, it has zero voltage drop and will carry any current imposed on it. When a switch is off, it blocks the flow of current regardless of the voltage across it. The device power, the product of the switch voltage and current, is identically zero at all times. A switch therefore controls energy flow with no loss. In addition, reliability is also high. Household light switches perform over decades of use and perhaps 100,000 operations. Unfortunately, a mechanical switch does not meet all practical needs. A switch in a power supply may function 100,000 times each second. Even the best mechanical switch will not last beyond a few million cycles. Semiconductor switches (without this limitation) are the devices of choice in power converters.

A circuit built from ideal switches will be lossless. As a result, switches are the main components of power converters, and many people equate power electronics with the study of switching power converters. Magnetic transformers and lossless storage elements such as capacitors and inductors are also valid components for use in power converters. The complete concept, shown in Fig. 1.3, represents a power electronic system. Such a system consists of an electrical energy source, an electrical load, a power electronic circuit, and a control function.

The power electronic circuit contains switches, lossless energy storage elements, and magnetic transformers. The controls take information from the source, the load, and the designer and then determine how the switches operate to achieve the desired conversion. The controls are built up with low-power analog and digital electronics.

Switching devices are selected based on their power handling rating—the product of their voltage and current ratings—rather than on power dissipation ratings. This contrasts with other applications of electronics, in which power dissipation ratings dominate. For instance, a typical stereo receiver performs a conversion from ac line input to audio output. Many audio amplifiers do not use the techniques of power electronics, and the semiconductor devices do not act as switches. A commercial 100 W amplifier, using a class A, B, or AB design, usually employs transistors big enough to dissipate the full 100 W, and the circuit is likely to be <10% energy efficient in normal use. In contrast, switching amplifiers, class D, E, or F, use the techniques of power electronics. They provide dramatic efficiency improvements. A home theater system implemented with switching amplifiers can exceed 90% energy efficiency in a small package. The amplifiers can even be packed inside the loudspeakers.

Switches can reach extreme power levels, far beyond what might be expected for a given size. Consider the following examples:

1.2.2 The Reliability Objective—Simplicity and Integration

High-power applications lead to interesting issues. In an inverter, the semiconductors often manipulate 30 times their power dissipation capability or more, which implies that only about 3% of the power being controlled is lost. A small design error, unexpected thermal problem, or minor change in layout could alter this somewhat. For instance, if the loss turns out to be 4% rather than 3%, the device stresses are 33% higher, and quick failure is likely to occur. The first issue for reliability in power electronic circuits is to keep device voltage, current, and power dissipation levels well within rating limits. This is challenging when power handling levels are high.

The second issue for reliability is simplicity. It is well established in electronics design that the more parts there are in a system, the more likely it is to fail. Power electronic circuits tend to have few parts, especially in the main energy flow paths. Necessary operations must be carried out through shrewd use of these parts. Often, this means that sophisticated control strategies are applied to seemingly simple conversion circuits.

The third issue for reliability is integration. One way to avoid the reliability-complexity trade-off is to integrate multiple components and functions together. A microprocessor, for example, might contain millions of gates. All interconnections and signals flow within a single chip, and the reliability is nearly that of a single part. Integrated power electronic modules [6] achieve this. Manufacturers package multiple switching devices, with their interconnections and protection components, together as a unit. Control circuits for converters are also integrated as much as possible to keep the reliability high. The package itself is a factor in reliability, and one that is a subject of active research. Many semiconductor packages include small bonding wires that can be susceptible to thermal or vibration damage. The small geometries also tend to enhance electromagnetic interference among the internal circuit components.

Electrical energy sources take the form of dc voltage sources at various values, sinusoidal ac sources and polyphase sources, among others. A power electronic circuit might be asked to transfer energy between two different dc voltage levels, between an ac source and a dc load or between sources at different frequencies. It might be used to adjust an output voltage or power level, drive a nonlinear load, or control a load current. In this section, a few basic converter arrangements are introduced, and energy conservation provides a tool for analysis.

Example 1.3

Consider the circuit shown in Fig. 1.4. It contains an ac source, a switch, and a resistive load. It is a simple power electronic system.

Let us assign a (somewhat arbitrary) control scheme to the switch. What if the switch is turned on whenever $V_{\mathrm{a}\mathrm{c}}>0$ and turned off otherwise? The input and output voltage waveforms are shown in Fig. 1.5. The input has a time average of 0 and root-mean-square (RMS) value equal to $V_{\mathrm{peak}}/\surd 2$, where V_peak is the maximum value of V_ac. The output has a nonzero average value given by

$\begin{array}{c}〈v_{\mathrm{out}}\left(t\right)〉=\frac{1}{2\pi }\left({\int }_{-\pi /2}^{\pi /2}{V}_{\mathrm{peak}}cos\theta \mathrm{d}\theta +{\int }_{\pi /2}^{3\pi /2}0\mathrm{d}\theta \right)\\ =\frac{V_{\mathrm{peak}}}{\pi }=0.3183V_{\mathrm{peak}}\end{array}$

and an RMS value equal to V_peak/2. Since the output has nonzero dc voltage content, the circuit can be used as an ac-dc converter. To make it more useful, a low-pass filter would be added between the output and the load to smooth out the ac portion. This filter needs to be lossless and will be constructed from only inductors and capacitors.

The circuit in Example 1.3 acts as a half-wave rectifier with a resistive load. With the hypothesized switch action, a diode can substitute for the ideal switch. The example confirms that a simple switching circuit can perform power conversion functions. But note that a diode is not, in general, the same as an ideal switch. A diode places restrictions on the current direction, but a true switch would not. An ideal switch allows control over whether it is on or off. In contrast, a diode's operation is constrained by circuit variables.

Consider a second half-wave circuit, now with a series $L-R$ load, shown in Fig. 1.6:

Example 1.4

A series diode $L-R$ circuit has ac voltage source input. This circuit operates much differently than the half-wave rectifier with resistive load. A diode will be on if forward-biased and off if reverse-biased. In this circuit, when the diode is off, the current will be zero. Whenever the diode is on, the circuit is the ac source with $L-R$ load. Let the ac voltage be V₀ cos(ωt). From Kirchhoff's voltage law (KVL),

$V_{0}cos\left(\omega t\right)=L\frac{\mathrm{d}i}{\mathrm{d}t}+Ri\text{.}$

Let us assume that the diode is initially off (this assumption is arbitrary, and we will check it as the example is solved). If the diode is off, the diode current is $i=0$, and the voltage across the diode will be v_ac. The diode will become forward-biased when v_ac becomes positive, so it will turn on when the input voltage makes a zero crossing in the positive direction. This allows us to establish initial conditions for the circuit, $i\left(t_0\right)=0,t_0=-\pi /\left(2\omega \right)$. The differential equation can be solved in a conventional way to give

$\begin{array}{c}i\left(t\right)=V_0\left[\frac{\omega L}{R^2+{\omega }^2{L}^2}exp\left(\frac{-t}{\tau }-\frac{\pi }{2\omega \tau }\right)\right.+\frac{R}{R^2+{\omega }^2{L}^2}cos\left(\omega t\right)\\ \left.+\frac{\omega L}{R^2+{\omega }^2{L}^2}sin\left(\omega t\right)\right]\end{array}$

  (1.2)

where τ is the time constant L/R. What about when the diode is turned off? The first guess might be that the diode turns off when the voltage becomes negative, which is not correct. From the solution, we can determine that the current is not zero when the voltage first becomes negative. If the switch attempts to turn off, it must instantly drop the inductor current to zero. The derivative of current in the inductor, di/dt, would become negative infinite. The inductor voltage L(di/dt) similarly becomes negative infinite, and the devices are destroyed. What really happens is that the falling current allows the inductor to maintain forward bias on the diode. The diode will turn off only when the current reaches zero. A diode has definite properties that determine the circuit action, and both the voltage and current are relevant. Fig. 1.7 shows the input and output waveforms for a time constant τ equal to about one-third of the ac waveform period.

1.4.2 The Method of Energy Balance

Any circuit must satisfy conservation of energy. In a lossless power electronic circuit, energy is delivered from source to load, possibly through an intermediate storage step. The energy flow must balance over time such that the energy drawn from the source matches that delivered to the load. The converter in Fig. 1.8 serves as an example of how the method of energy balance can be used to analyze circuit operation.

If the inductor in the polarity reversal circuit is moved instead to the input, a step-up function is obtained. Consider the circuit shown in Fig. 1.9 in the following example.

The circuits in the preceding examples have few components, provide useful conversion functions, and are efficient. If the switching devices are ideal, each circuit is lossless. Over the history of power electronics, development has tended to flow around the discovery of such circuits: a circuit with a particular conversion function is discovered, analyzed, and applied. As the circuit moves from laboratory testing to a complete commercial product, control and protection functions are added. The power portion of the circuit remains close to the original idea. The natural question arises as to whether a systematic approach to conversion is possible: can we start with a desired function and design an appropriate converter, rather than starting from the converter and working backward toward the application? What underlying principles can be applied to design and analysis? In this chapter, a few of the key concepts are introduced. Note that, although many of the circuits look deceptively simple, all these circuits are nonlinear systems with unusual behavior.

The most readily apparent difference between a power electronic circuit and other types of electronic circuits is the switch action. In contrast to a digital circuit, the switches do not indicate a logic level. Control is effected by determining the times at which switches should operate. Whether there is just one switch or a large group, there is a complexity limit: if a converter has m inputs and n outputs, even the densest possible collection of switches would have a single switch between each input and output line. The $m\times n$ switches in the circuit can be arranged according to their connections. The pattern suggests a matrix, as shown in Fig. 1.10.

Power electronic circuits fall into two broad classes [2]:

1. Direct switch matrix circuits. In these circuits, energy storage elements are connected to the matrix only at the input and output terminals. The storage elements effectively become part of the source or the load. In the literature, ac-ac versions of these circuits are sometimes called matrix converters.

2. Indirect switch matrix circuits, also termed embedded converters. These circuits, like the polarity-reverser example, have energy storage elements connected within the matrix structure. Indirect switch matrix circuits are most commonly analyzed as a cascade connection of direct switch matrix circuits with storage in between.

The switch matrices in realistic applications are small. A 2×2 switch matrix, for example, covers all possible cases with a single-port input source and a two-terminal load. The matrix is commonly drawn as the H-bridge shown in Fig. 1.11. A three-phase bridge rectifier is shown in Fig. 1.12. There are three possible inputs, and the two terminals of the dc circuit provide outputs, which gives a 3×2 switch matrix. In a computer power supply with five separate dc loads, the switch matrix could be 2×10. Very few practical converters have more than 24 switches, and most designs use fewer than 12.

A switch matrix provides a way to organize devices for a given application. It also helps us focus on three major task areas, which must be addressed individually and effectively in order to produce a useful power electronic system:

• The “hardware” task—Build a switch matrix. This involves the selection of appropriate semiconductor switches and the auxiliary elements that drive and protect them.

• The “software” task—Operate the matrix to achieve the desired conversion. All operational decisions are implemented by adjusting switch timing.

• The “interface” task—Add energy storage elements to provide the filters or intermediate storage necessary to meet the application requirements. Lossless filters with simple structures are required.

In a rectifier or other converter, we must choose the electronic parts, how to operate them, and how best to filter the output to satisfy the needs of the load.

1.5.2 Implications of Kirchhoff's Voltage and Current Laws

A major challenge of switch circuits is their capacity to “violate” circuit laws. First, consider the simple circuits shown in Fig. 1.13. We might try the circuit shown in Fig. 1.13A for ac-dc conversion, but there is a problem. According to Kirchhoff's voltage law (KVL), the “sum of voltage drops around a closed loop is zero.” However, with the switch closed, the sum of voltages around the loop is not zero. In reality, this is not a valid result. Instead, a very large current will flow and cause a large $I\cdot R$ drop in the wires. KVL will be satisfied by the wire voltage drop, but a fire or, better yet, fuse action, might result. There is, however, nothing that would prevent an operator from trying to close the switch. KVL, then, implies a crucial restriction: a switch matrix must not attempt to interconnect unequal voltage sources directly. Notice that a wire or dead short can be thought of as a voltage source with $V=0$, so KVL is a generalization of avoiding shorts across an individual voltage source.

A similar constraint holds for Kirchhoff's current law (KCL) that states that “currents into a node must sum to zero.” When current sources are present in a converter, we must avoid any attempts to violate KCL. In Fig. 1.13B, if the current sources are different and if the switch is opened, the sum of the currents into the node will not be zero. In a real circuit, high voltages will build up and cause an arc to create another current path. This situation has real potential for damage, and a fuse will not help. As a result, KCL implies the restriction that a switch matrix must not attempt to interconnect unequal current sources directly. An open circuit can be thought of as a current source with $I=0$, so KCL applies to the problem of opening an individual current source.

In contrast to conventional circuits, in which KVL and KCL are automatically satisfied, switches do not “know” KVL or KCL. If a designer forgets to check and accidentally shorts two voltages or breaks a current source connection, some problem or damage will result. KVL and KCL place necessary constraints on the operation of a switch matrix. In the case of voltage sources, switches must not act to create short-circuit paths among unlike sources. In the case of KCL, switches must act to provide a path for currents. These constraints drastically reduce the number of valid switch-operating conditions in a switch matrix, thereby leading to manageable operating design problems.

When energy storage is included, there are interesting implications of the circuit law restrictions. Fig. 1.14 shows two “circuit law problems.” In Fig. 1.14A, the voltage source will cause the inductor current to ramp up indefinitely, since $V=L\mathrm{d}i/\mathrm{d}t$. We might consider this to be a “KVL problem” because the long-term effect is similar to shorting the source. In Fig. 1.14B, the current source will cause the capacitor voltage to ramp toward infinity. This causes a “KCL problem”; eventually, an arc will be formed to create an additional current path, just as if the current source had been opened. However, these connections are not problematic if they are only temporary. It should be evident that an inductor will not support dc voltage, and a capacitor will not support dc current. On average, over an extended time interval, the voltage across an inductor must be zero, and the current into a capacitor must be zero.

1.5.3 Resolving the Hardware Problem—Semiconductor Devices

A switch is either on or off. When on, an ideal switch will carry any current in any direction. When off, it will never carry current, no matter what voltage is applied. It is entirely lossless and changes from its on-state to its off-state instantaneously. A real switch can only approximate an ideal switch. The following are the aspects of real switches that differ from the ideal:

• Limits on the amount and direction of on-state current.

• A nonzero on-state voltage drop (such as a diode forward voltage).

• Some levels of leakage current when the device is supposed to be off.

• Limitations on the voltage that can be applied when off.

• Operating speed. The duration of transition between the on-states and off-states is important.

The degree to which the properties of an ideal switch must be met by a real switch depends on the application. For example, a diode can easily be used to conduct dc current; the fact that it conducts only in one direction is often an advantage, not a weakness.

Many different types of semiconductors have been applied in power electronics. In general, these fall into three groups:

1. Diodes, which are used in rectifiers, dc-dc converters, and in a wide range of supporting roles.

2. Transistors, which in general are suitable for control of single-polarity circuits. Several types of transistors are applied to power converters. The insulated-gate bipolar transistor (IGBT) type is unique to power electronics and has good characteristics for applications such as inverters.

3. Thyristors, which are multijunction semiconductor devices with latching behavior. In general, thyristors can be switched with short pulses and then maintain their state until current is removed. The characteristics are especially well suited to high-power controllable rectifiers, but they have been applied to all power conversion applications.

Some of the features of the most common power semiconductors are listed in Table 1.1. The table shows a wide variety of speeds and rating levels. As a rule, faster speeds apply to lower ratings. For each device type, cost tends to increase both for faster devices and for devices with higher power handling capacity.

Conducting direction and blocking behavior are fundamentally tied to the device type, and these basic characteristics constrain the choice of device for a given conversion function. Consider again a diode. It carries current in only one direction and always blocks current in the other direction. Ideally, the diode exhibits no forward voltage drop or off-state leakage current. Although an ideal diode lacks the many features of an ideal switch, it is an important switching device. Other real devices operate with polarity limits on current and voltage and have corresponding ideal counterparts. It is convenient to define a special type of switch to represent this behavior, the restricted switch:

The diode always permits current flow in one direction, while blocking flow in the other direction. It therefore represents a forward-conducting reverse-blocking restricted switch and operates in one quadrant on a graph of device current versus voltage. This unipolar function is automatic; the two diode terminals provide all the necessary information for switch action. Other restricted switches require a third gate terminal to determine their state. Consider the polarity possibilities given in Table 1.2. Additional functions such as bidirectional-conducting reverse-blocking can be obtained by reverse connection of one of the five types in the table.

Table 1.2

The types of restricted switches

Action	Device	Quadrants	Restricted switch symbol	Device symbol
Carries current in one direction, blocks in the other (forward-conducting reverse-blocking)	Diode
Carries or blocks current in one direction (forward-conducting forward-blocking)	BJT, IGBT
Carries in one direction or blocks in both directions (forward-conducting bidirectional-blocking)	GTO
Carries in both directions but blocks only in one direction (bidirectional-conducting forward-blocking)	FET
Fully bidirectional	Ideal switch

The quadrant operation shown in the table indicates polarities. For example, the current in a diode will be positive when on, and the voltage will be negative when off. This means diode operation is restricted to the single quadrant comprising the upper vertical (current) axis and the left horizontal (voltage) axis. Other combinations appear in the table. Symbols for restricted switches can be built up by interpreting the diode's triangle as the current-carrying direction and the bar as the blocking direction. Five types of symbols can be drawn as shown in Table 1.2. These symbols are used infrequently but are useful for showing the polarity behavior of switching devices. A circuit drawn with restricted switches represents an idealized power converter.

Restricted switch concepts guide the selection of devices. For example, consider an inverter intended to deliver ac load current from a dc voltage source. A switch matrix built to perform this function must be able to manipulate ac current and dc voltage. Regardless of the physical arrangement of the matrix, we would expect bidirectional-conducting forward-blocking switches to be useful for this conversion. This is a correct result: modern inverters operating from dc voltage sources are built with field effect transistors (FETs) or with IGBTs packaged with reverse-parallel diodes. As new power devices are introduced to the market, it is straightforward to determine what types of converters will use them.

1.5.4 Resolving the Software Problem—Switching Functions

The physical $m\times n$ switch matrix can be associated with a mathematical $m\times n$ switch state matrix. Each element of this matrix, called a switching function, shows whether the corresponding physical device is on or off:

Fig. 1.15 shows a typical switching function. It is periodic, with period T, representing the likely repetitive switch action in a power converter. For convenience, it is drawn on a relative time scale that begins at 0 and draws out the square wave period by period. The actual timing is arbitrary, so the center of the first pulse is defined as a specified time t₀ in the figure. In many converters, the switching function is generated as an actual control voltage signal that might drive the gate of an FET or IGBT.

The timing of switch action is the only alternative for control of a power converter. Because switch action can be represented with a discrete-valued switching function, timing can be represented within the switching function framework. On the basis of Fig. 1.15, a generic switching function can be characterized completely with three parameters:

1. The duty ratio, D, is the fraction of time during which the switch is on. For control purposes, the on-time interval or pulse width can be adjusted to achieve a desired result. We can term this adjustment process pulse-width modulation (PWM), perhaps the most important process for implementing control in power converters.

2. The frequency $f_{\mathrm{switch}}=1/T$ (with radian frequency $\omega =2\pi f_{\mathrm{switch}}$) is most often constant, although not in all applications. For control purposes, frequency can be adjusted. This strategy is sometimes used in low-powerdc-dc converters to manage wide load ranges. In other converters, frequency control is unusual because the operating frequency is often dictated by the application.

3. The time delay t₀ or phase ${\phi }_0=\omega t_0$. Rectifiers often use phase control to provide a range of adjustment. Phase-shifted bridge circuits are common for high-power dc-dc conversion. A few specialized ac-ac converter applications use phase modulation.

With just three parameters to vary, there are relatively few possible ways to control any power electronic circuit. Inverters and dc-dc converters usually rely on duty ratio adjustment (PWM) to alter their behavior. Phase control is common in controlled rectifier applications.

Switching functions are powerful tools for the general representation of converter action [7]. The most widely used control approaches derive from averages of switching functions [2,8]. Their utility comes from their application in writing circuit equations. For example, in the boost converter shown in Fig. 1.9, the loop and node equations change depending on which switch is acting at a given moment. The two possible circuit configurations each have distinct equations. Switching functions allow them to be combined. By assigning switching functions q₁(t) and q₂(t) to the left- and right-switching devices, respectively, we obtain

$\begin{array}{l}{q}_1\left(V_{\mathrm{in}}-L\frac{\mathrm{d}{i}_L}{\mathrm{d}t}=0\right),\\ q_1\left(C\frac{\mathrm{d}{v}_C}{\mathrm{d}t}+\frac{v_C}{R}=0\right),\mathrm{left}\mathrm{switch}\mathrm{on}\end{array}$

$\begin{array}{l}{q}_2\left(V_{\mathrm{in}}-L\frac{\mathrm{d}{i}_L}{\mathrm{d}t}=v_C\right),\\ q_2\left(C\frac{\mathrm{d}{v}_C}{\mathrm{d}t}+\frac{v_C}{R}=i_L\right),\mathrm{right}\mathrm{switch}\mathrm{on}\end{array}$

Because the switches alternate and the switching functions must be 0 or 1, these sets of equations can be combined to give

$V_{\mathrm{in}}-L\frac{\mathrm{d}{i}_{\mathrm{L}}}{\mathrm{d}t}=q_2{v}_{\mathrm{C}},C\frac{\mathrm{d}{v}_{\mathrm{C}}}{\mathrm{d}t}+\frac{v_{\mathrm{C}}}{R}=q_2{i}_{\mathrm{L}}$

The combined expressions are simpler and easier to analyze than the original equations.

For control purposes, the average of equations such as Eq. (1.10) often proceeds with the replacement of switching functions q with duty ratios d. The discrete-time action of a switching function thus will be represented by an average duty ratio parameter. Switching functions, advantages gained by averaging, and control approaches such as PWM are discussed at length in several chapters in this handbook.

1.5.5 Resolving the Interface Problem—Lossless Filter Design

Lossless filters for power electronic applications are sometimes called smoothing filters [9]. In applications in which dc outputs are of interest, such filters are commonly implemented as simple low-pass LC structures. The analysis is facilitated because in most cases the residual output waveform, termed ripple, has a predictable shape. Filter design for rectifiers or dc-dc converters is a question of choosing storage elements large enough to keep ripple low, but not so large that the whole circuit becomes unwieldy or expensive.

Filter design is more challenging when ac outputs are desired. In some cases, this is again an issue of low-pass filter design. However, in many applications, low-pass filters are not adequate to meet low noise requirements. In these situations, active filters can be used. In power electronics, the term active filter refers to lossless switching converters that actively inject or remove energy moment by moment to compensate for distortion. The circuits (discussed in Chapter 41—active filters in this handbook) are not related to the linear active filter op-amp circuits used in analog signal processing. In ac applications, there is a continuing opportunity for innovation in filter design.